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Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

Authors Gramoz Goranci, Monika Henzinger, Pan Peng

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ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic graph algorithms
  • effective resistance
  • separable graphs
  • Schur complement
  • conditional lower bounds

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Abstract

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c<1. We give a fully dynamic algorithm that maintains (1+epsilon)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with O~(sqrt{n}/epsilon^2) worst-case update time and O~(sqrt{n}/epsilon^2) worst-case query time, if G is guaranteed to be sqrt{n}-separable (i.e., it is taken from a class satisfying a sqrt{n}-separator theorem) and its separator can be computed in O~(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest.
We complement our result by proving that for any two fixed vertices s and t, no incremental or decremental algorithm can maintain the s-t effective resistance for sqrt{n}-separable graphs with worst-case update time O(n^{1/2-delta}) and query time O(n^{1-delta}) for any delta>0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false.
We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false.

Cite As Get BibTex

Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.40

Author Details

Gramoz Goranci
  • University of Vienna, Faculty of Computer Science, Vienna, Austria
Monika Henzinger
  • University of Vienna, Faculty of Computer Science, Vienna, Austria
Pan Peng
  • Department of Computer Science, University of Sheffield, Sheffield, UK

Funding

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506.

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